I am following this tutorial for Autumn Leaves on youtube:. Around timestamp 3:40, the presenter plays the notes F - A - Bb - D and calls it a Gm7 chord. This baffled me immensely. At first I swore there was just a mistake in the video. A quick google search tells me this is a 'mediant' Gm7th chord. What is the reason for calling this chord Gm7th instead of a Bb7, especially when the piece already has a ton of Bb7's? I'm sure there's some music theoretic explanation that I have not encountered before, but I don't know, this seems like particularly egregious notational torture. How am I ever supposed to know whether I should be playing a 'mediant' or 'tonic' or 'subtonic' version of a chord from a chord sheet?
Yes, it is Gm7 (or Gm9) with missing G.
This is called rootless voicing, and it is a common way of playing chords in piano jazz arrangements. It is assumed that the root is played by some other instrument, like bass. It doesn't need to be doubled, and if doubled it may even clash.
Please note that the rootless version of the chord still contains the two most important tensions: the third and the seventh, so by playing this way you still preserve the large part of the harmonic motion. It also gives you more freedom in adding upper structures, like 9 in this example.
The video is misleading at this point, because the left hand (or bass part) has been left out. The left hand (or bass) would be playing
G, so the total chord would be
G-Bb-D-F-A, which, strictly speaking, is a
Gm9. The added 9th, the
A is a "safe" addition in this situation and adds nice color to the chord.
It's not a
Bb7, because that chord has
Ab rather than
A. As played, without the root of the
Gm7 chord, it would be
The whole verse is made up from 2-5-1s, as stated in the clip. So it's not surprising that after Am7♭5 - D7 - it's Gm, of some sort - rather than B♭, of some sort. Were it B♭, the preceding chord would more likely be F(7).
The fact that the chord shown is in 3rd inversion is really neither here nor there. The bassist at that point would most likely play a G, at least, leaving the piano left hand to play whatever other notes of the chord were fancied. A choice of G, B♭, D, F are obvious, with the A making Gm9, an extension of the original.
And even if there was no bassist playing, there's nothing in law saying chords must be in root position - quite often in jazz that's not the case, which is one reason why jazz voicings sound like, well, jazz.